University of Illinois at Urbana–Champaign
Topic: Heron's Formula for Determining the Area of a Triangle:
Introduction:
The traditional introduction to the area of a triangle involves the formula:
Area = [pic]
This formula is easiest when you have right angle triangles or those whose heights can be determined.
What if we don't know the heights, or bases, but we do know all three sides???
We can still determine the area!
|This formula is attributed to Heron of Alexandria but can be traced to Archimedes. |
|It involves two separate calculations: |
|1st: [pic] (where a,b,c are sides of a triangle) |
|2nd: AREA = [pic] |
Activity: Research Heron (or Hero) of Alexandria and write a paragraph about his other accomplishments. Use whatever resources available. Here is a start:
Hero of Alexandria, a Greek mathematician and scientist. His name is also spelled Heron. Heron (Hero) is credited with developing the formula for finding the area of a triangle in terms of its sides, but this formula was probably developed before his time (by Archimedes).
Here is an Example: Take a Triangle with sides in inches (3, 4, 5) – A Pythagorean Triple in fact.
[pic]Area = [pic] = 6 inches2
Using Area = [pic] We get ½ * 4 * 3 = 6 inches2
Or we could use the calculator program (screenshots):
| | | |
Now, what if we enter measurements and get a result like the following?
| | |
This is not a “real” triangle as the sides do not fulfill the requirements of the triangle inequality theorem.
|The Triangle Inequality Theorem states the following: |
| |
|The sum of the lengths of any two sides of a triangle is greater than the length of the third side. So... |
|[pic], [pic], and [pic] |
in our case we have
17 + 2 > 4 and 17 + 4 > 2, but 4 + 2 < 17 so these sides fail the triangle inequality theorem. An example of this type of triangle is depicted below. Note that the segments do not unite to form a triangle.
To demonstrate this we can do the following:
Take a piece of twine and divide it into 17 inches, 2 inches, and 4 inches. This is shown here:
[pic]
Then take those pieces of twine and attempt to form a triangle, shown here:
[pic][pic]
Note that in either case a triangle cannot be formed with the lengths of twine provided.
Activity: Repeat the above activity and create a triangle using the lengths of twine provided, measure the lengths and record them in the table below. This may look something like the photo below (it is not required to look like this though).
[pic]
Activity: Using Heron's Formula determine the areas of triangles with the following side measurements (in centimeters).
1. (5, 12, 16)
2. (12, 44, 23)
3. (61, 2, 14)
4. (12, 11, 12)
5. (20, 24, 11)
6. (18, 7, 21)
You may also want to draw them out to see how they would look.
Activity: Check them with the online tools listed below or the calculator program to see if you are correct.
Activity: Are all the “triangles” presented really triangles? If not, show how they do not fit the triangle inequality theorem.
[pic]Resources for Additonal Exploration:
Jim Dildine has created an online, interactive version of these topics. This is located at
mste.uiuc.edu/dildine/heron/triarea.html
The calculator program file is located there or a direct link can be accessed from
mste.uiuc.edu/dildine/heron/heron2.8xp
Jim has also included an Excel file, a Geometer's Sketchpad file, and an interactive Java Sketch. Available here:
mste.uiuc.edu/dildine/heron/triarea.html
[pic]
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